Properties

Label 2.720.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $720$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.2880.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.20.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} - 4x^{2} + 2x + 13 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 23\cdot 41 + 12\cdot 41^{2} + 21\cdot 41^{3} + 4\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 19 + 8\cdot 41 + 35\cdot 41^{2} + 37\cdot 41^{3} + 11\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 9\cdot 41 + 6\cdot 41^{2} + 20\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 30 + 40\cdot 41 + 27\cdot 41^{2} + 2\cdot 41^{3} + 18\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,3)(2,4)$
$(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)$$-2$
$2$$2$$(1,3)(2,4)$$0$
$2$$2$$(1,2)$$0$
$2$$4$$(1,4,2,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.